Abstract
<title>Abstract</title> <p>This paper presents a systematic bifurcation analysis of a rear-wheel-drive vehicle, modelled through three state variables (longitudinal velocity, lateral velocity, and yaw rate) using the Modified Elliptical Method (MEM) to represent combined-slip tyre forces. The MEM is a closed-form, differentiable tyre model that captures the elliptical friction constraint linking longitudinal and lateral force generation, and it admits direct Jacobian computation for eigenvalue analysis. Because dynamic load transfer couples the normal forces to longitudinal acceleration, the resulting equations of motion form an implicit ordinary differential equation system. Stability is therefore assessed through an effective Jacobian that corrects the conventional state Jacobian for its dependence on state derivatives, a correction shown to be essential for correctly classifying stability near the handling limit. Equilibrium points are located using a hybrid procedure that combines a genetic algorithm for global search with Newton--Raphson refinement for local accuracy. The resulting bifurcation portrait, mapped across both rear slip ratio and front steering angle, reveals three qualitatively distinct operating regimes: stable handling, steering-direction drifting, and counter-steering drifting. In the steering-direction regime, a supercritical Hopf bifurcation marks the onset of a stable limit cycle whose amplitude grows with distance from the critical slip ratio; at sufficiently large steering angles, a second Hopf bifurcation renders the drifting equilibrium oscillatorily unstable as well. In the counter-steering regime, fold bifurcations set the existence boundaries of the drifting equilibria, and these fold curves coalesce at a codimension-two cusp bifurcation, confirmed analytically by a non-zero cubic normal-form coefficient. The cusp organises a bistable region bounded by two coalescing fold curves, giving rise to hysteresis and the potential for catastrophic, discontinuous transitions between cornering equilibria. Together, these results provide stability boundaries directly relevant to electronic stability control design and to safety-envelope specification for autonomous vehicle controllers.</p>